No, it would not.
One starting point (in fact a postulate) of special relativity is
that the speed of light in vacuum is the same for every observer,
regardless of the motion of them or the source. Sound does not
fulfil this requirement for a number of reasons:
- Sound
waves need a medium in which to propagate. The speed is
different depending on the medium, and there is no sound in
space. Even with air it varies according to pressure and
humidity, though it is roughly constant for many practical
purposes (about 330 m/s).
- The speed of sound is fixed relative to the
medium. Therefore, the observed speed depends on the motion of the
medium (e.g. wind).
The invariance of the speed of light relates to the lack of a
required medium, and it is not unique to light. It applies to all
forms of electric and magnetic fields (light being a special
case), the strong nuclear force, and supposedly
gravitation. However, it is the same c in every case; relativity
would collapse if we found something with an invariant speed
different from c. It would be interesting, though.
Considering Pender's writeup, I might add that evolution has
produced vision several times independently throughout the
history of Earth. It is expected to happen in a world dominated
by light.
On a final note, you could imagine that the people discover or
invent something that is faster than the speed of sound in air. For
instance, the strings in m_turner's example may have a rather high
speed of sound along them. It is twice the length times the
sound frequency, from some very basic wave physics. I believe
the people would know this, if they were ready to develop
relativity. If you take something like the string in a properly
tuned guitar, you get wave speeds in considerable excess of 330 m/s.